The Fibonacci identities of orthogonality

In even dimensions, the orthogonal projection onto the two dimensional space of second order recurrence sequences is particularly nice: it is a scaled Hankel matrix whose entries consist of the classical Fibonacci sequence. A new proof is given of this result, and new Fibonacci identities are derive...

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Bibliographic Details
Published in:Linear algebra and its applications 2015-06, Vol.475, p.80-89
Main Authors: Hawkins, Kyle, Hebert-Johnson, Ursula, Mathes, Ben
Format: Article
Language:English
Subjects:
Fibonacci identities
Fibonacci numbers
Orthogonal projection
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Summary:In even dimensions, the orthogonal projection onto the two dimensional space of second order recurrence sequences is particularly nice: it is a scaled Hankel matrix whose entries consist of the classical Fibonacci sequence. A new proof is given of this result, and new Fibonacci identities are derived from it. Examples are given showing that familiar Fibonacci identities can be viewed as special cases. We show that the projection in odd dimensions can be written as a rank one Lucas perturbation of a scaled Lucas Hankel matrix, from which more Fibonacci identities are derived.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2015.01.024